Fig. 1

Graph representation of multipartite CV entangled states and their preparation. a CV two-mode Gaussian entangled state. b Three-mode GHZ state. c Four-mode square Gaussian cluster state, respectively. d, e show the beam-splitter network used to generate the three-mode GHZ state and four-mode square Gaussian cluster state, respectively. The phase shift (PS) is realized by locking the relative phase of two light beams at the corresponding beam splitter. f–h show the schematics of preparation and measuring the two-mode Gaussian entangled state, three-mode GHZ state, and four-mode square Gaussian cluster state, respectively. PS phase shift, NOPA nondegenerate optical parametric amplifier, HWP half-wave plate, PBS polarizing beam splitter, LO local oscillator, HD1–4 homodyne detectors, DM dichroic mirror